Question

Solve the questions

Answer 1

Let's do the trig part first, then the calculus.  First we simplify the integrand.

$\cot^3 x \sin ^3 x=\dfrac{\cos^3 x}{\sin^3 x} \sin^3 x= \cos^3 x$

The triple angle formula is:

$\cos 3x = 4 \cos^3 x - 3 \cos x$

or

$\cos^3 x = \frac 1 4 (\cos 3x + 3 \cos x)$

Now we can integrate:

$\displaystyle \int \cot^3 x \sin^3 x \ dx = \int \cos^3 x\ dx= \int \tfrac 1 4 (\cos 3x + 3 \cos x)\ dx$

$=\frac 1 4 (\frac 1 3 \sin 3x + 3 \sin x) + C$

$=\frac 1 {12} \sin 3x + \frac 3 4 \sin x + C$